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Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chap. VII: Sous-algèbres de Cartan, éléments réguliers. Chap. VIII: Algèbres de Lie semi-simples déployées. (French) Zbl 0329.17002

Actualités Scientifiques et Industrielles, 1364. Paris: Hermann. 271 p. F 78.00 (1975).
(For chapters 1–6 see Zbl 0213.04103; Zbl 0244.22007; Zbl 0249.22001.)
This volume contains two more chapters of Bourbaki’s work on Lie groups and Lie algebras. Chapter 7 begins with a description of the weight space decomposition relative to a linear representation; introduces regular elements and uses them to construct Cartan subalgebras; and proves conjugacy by Zariski topology arguments. Next regular elements of a Lie group are defined and related to the Lie algebra. There is a section on Mal’cev’s concept of splittability (Jacobson’s term is “almost algebraic”), with the interesting result that a Lie algebra is splittable if and only if it has a splittable Cartan subalgebra.
Chapter 8 starts by looking at representations of \(sl(2,k)\), defines root systems, and proves existence of semisimple algebras with a given root system by Serre’s generator-relation method. Next comes the representation theory of semisimple algebras, a description of the representation ring; Weyl’s formula; maximal subalgebras of semisimple algebras; nilpotent elements; Chevalley orders; a detailed study of the classical algebras; and finally some tables.
The exposition is solid and, as usual, carried out with as great generality as possible. Bourbaki’s usual terminological flights of fancy are largely absent, which is to be applauded. A number of treatments of this material are now available: this one is no better than, and no worse than, most of them.

MSC:

17-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras
22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
22E60 Lie algebras of Lie groups